More than 50 years ago, G. Polya described the main parts of solving math problems in his book

**. These same steps help with the solutions of physics problem.**

*How to Solve It***1. First, make sure you really understand the problem. You should be able to paraphrase the problem without adding any new assumptions or removing any of the constraints of the original problem. Identify all the quantities you are given or asked for and name them with variables. You should also make a sketch to help you understand (and explain) how things relate to each other geometrically.**

**2. Next, devise a plan. This means identifying the equations and principles you will need to solve the problem. In most cases, you will have to use more than one equation, with the output of one equation becoming the input of another. Make sure you see these relationships.**

**3. Once you have reduced the physics problem to a math problem, use your math skills to solve it. Your math prerequisites should be enough for you to do this, but the lectures and textbook will contain important refreshers and explanations for techniques you may not have quite mastered. Even though you are probably not used to doing so, it is important to keep all your units at each step. This will help you in two main ways.**

**a) It makes you think about the numbers you are using. In introductory physics, we use the acceleration 9.8 m/s**

^{2}very frequently, but the velocity 9.8 m/s will only appear by chance. If you see the velocity 9.8 m/s in your calculations, you are probably making a mistake, so double-check. You won't notice this if you only write 9.8.**b) It performs the same check for numerical problems as dimensional analysis does for algebraic problems.**

**4. Having carried out the math, the problem is solved, but it is still important to look back at your results and perform any checks you can.**

**a) Sometimes you may see that the answer is obviously wrong. For example, a test several years ago asked how long it would take a diver off a 10-meter platform to hit the water, and a student answered 300 seconds. That's five minutes to fall about thirty feet! An answer like that is obviously wrong.**

**b) Check to see whether your answer satisfies the constraints and conditions of the problem. For example, the quadratic equation you must solve for the diver problem above has two solutions: 1.43 seconds before the diver steps off the platform or 1.43 seconds after the diver steps off the platform. However, the diver did not shoot up through the surface of the water to the level of the platform; before the dive, she was just standing on the platform, not in free fall. Only the second answer meets the conditions of the problem.**

**c) If your problem has a numerical answer, you could make an order-of-magnitude estimate to see if the answer you give is in the right ballpark. These are most important if you are dealing with a problem involving very large numbers (like are found in problems relating to the planets) or very small numbers (as are found in problems involving atoms or electrons).**

**d) If your problem requires an algebraic answer, you could perform dimensional analysis. Dimensional analysis can't tell you that your answer is right, but it can sometimes tell you that your answer is wrong.**

The problem-solving sheets will help guide you through these steps. In practice, the greatest difficulty most students have is with either step 1 or step 2, so let's look at those in more detail.

## Understand the Problem

Read through the problem once to get the general idea, but then go back and look for keywords.

**Example 1:**Spider-Man falls 10.0 m off a building ledge before a strand of his web starts to slow him. The strand of web brings him to a stop in 2.0 m. What is the acceleration (assumed constant) as the web catches Spidey?

In this example, there are 3 key positions: the top of the building, where Spider-Man starts falling from rest; the point 10.0 m below that where the web starts to stretch; and the point 2.0 m below that (12.0 m) below the top of the building where he again comes to rest.

Since Spidey is just falling off the building, his initial vertical velocity will be zero when he is at the top of the building. For the first 10.0 m he is freely falling, so his (vertical) acceleration is a constant -9.8 m/s

^{2}over that distance. For the last 2.0 m he again has a constant acceleration (because both gravity, which continues to exert a force on him, and the tension in the web are both constant – not a realistic assumption for the web, as a matter of fact), but here the acceleration is not given. It is what we are looking for.

Notice how useful a sketch would be in keeping this straight in your mind! Such a sketch does need to show (not necessarily to scale) the two distances through which Spidey falls, but it does not need to contain elaborate artistic details. A stick figure for Spider-Man is just right!

If sketches are useful for one-dimensional problems, they are

*indispensable*for two-dimensional problems.

**Example 2**: A spring-loaded toy cannon shoots a small metal ball off the top of a table with an initial speed of 4.75 m/s. The ball is fired at an angle of 60

^{o}from the horizon at an initial height of 1.10 m. How far does the ball move horizontally by the time it hits the floor?

The hint for how we go about this problem comes from the wording of that last sentence. The solution has two main parts.

**(1) “How far does the ball move horizontally by [a certain] time?”**

*“How far” is the final result we are looking for, but to get it we need first to find out how much time passes until we stop measuring. That part comes from the second question:*

**(2) “At what time does the ball hit the floor?”**

As for the sketch(es), in this case you need to show the parabolic trajectory of the ball as it leaves the cannon. The initial height needs to be shown, as well as the horizontal distance you are seeking and the initial velocity vector. You should also show (probably in an expanded view) a right triangle showing the initial velocity vector together with its horizontal and vertical components.

## Devise a Plan

**1. Write out all the equations you think might be useful in the solution.**

**2. Circle every variable that you already know.**

**3. Draw a box around the final variable you are looking for.**

**4. If you find one equation in which all the variables are either circled or boxed, you can use that equation to solve the problem.**

**5. If not, see if you have a pair of equations in which**

**a) in one equation, everything is circled but one variable**

**b) in the other equation, everything is circled or boxed but one variable, and**

**c) the unmarked variable is the same in both a) and b).**

**6. Draw a parallelogram around the unmarked variable from step 5. This is your intermediate result.**

**7. Find the intermediate result from the equation that has everything circled except for the variable in the parallelogram.**

**8. Use value of the intermediate result in the other equation to find your final result.**

That sounds terribly complicated, but the examples already given (the Spider-Man problem and the projectile problem) should help clarify what I mean.

Example 1: First some definitions. Let's call the top of the building point A, the point 10.0 m below that where the web first starts to stretch B, and the point 2.0 m below that where Spidey stops again point C. Some quantities are defined at these points and will use a subscript A, B, or C; others are defined over the interval AB or the interval BC, so they will use those subscripts. Both intervals involve constant acceleration, so we can write out our equations for constant acceleration.

1. Δy

_{AB}= v

_{A}t

_{AB}+ ½ a

_{AB}t

_{AB}

^{2}

2. v

_{B}= v

_{A}+ a

_{AB}t

_{AB}

**3.**v

_{B}

^{2}= v

_{A}

^{2}+ 2 a

_{AB}Δy

_{AB}

4. Δy

_{BC}= v

_{B}t

_{BC}+ ½ a

_{BC}t

_{BC}

^{2}

5. v

_{C}= v

_{B}+ a

_{BC}t

_{BC}

6. v

_{C}

^{2}= v

_{B}

^{2}+ 2 a

_{BC}Δy

_{BC}

The only things we know are

Δy

_{AB}= -10.0 m,

v

_{A}= 0 m/s,

a

_{AB}= -9.8 m/s

^{2},

Δy

_{BC}= -2.0 m, and

v

_{C}= 0 m/s,

so draw circles around those variables. We are looking for a

_{BC}, so draw a box around that. None of the equations have all the variables in circles or boxes, so we follow Step 5 and look for a pair of equations that share one unmarked variable. Equations 3 and 6 satisfy the requirements, with v

_{B}as the shared unmarked variable. Draw a parallelogram around it; it is an intermediate result we will need. Equation 3 now has only circles and the parallelogram; it can be solved for v

_{B}, then Equation 6 can be solved for a

_{BC}.

Example 2: To simplify things, I'll assume you have already calculated the x- and y-components of the initial velocity. In the x-direction you have constant velocity, and in the y-direction you have constant acceleration, so the equations you might use are

**1.**Δy = v

_{iy}t + ½ a

_{y}t

^{2}

2. v

_{fy}= v

_{iy}+ a

_{y}t

3. v

_{fy}

^{2}= v

_{iy}

^{2}+ 2 a

_{y}Δy

**4.**Δx = v

_{ix}t

We know

v

_{ix}= (4.75 m/s) cos 60

^{o}= 2.375 m/s,

v

_{iy}= (4.75 m/s) sin 60

^{o}= 4.1136 m/s,

Δy = -1.1 m, and

a

_{y}= -9.8 m/s

^{2},

so draw circles around those variables. We are looking for Δx, so draw a box around that. Once again, when we come to Step 4, there are no equations that have only circled and boxed variables. However, Equations 1 and 4 are a pair satisfying the requirements of Step 5, with t as the intermediate result, so draw a parallelogram around t. At this point you should be able to see that you can solve Equation 1 for t and then substitute that intermediate value into Equation 4 to find Δx.

Of course, more complicated problems may require you to chain together three equations connected by two intermediate results, four equations connected by three intermediate results, or whatever. For example, the problem in Example 1 could have given Spider-Man's mass and asked for the net force on him while the web is slowing him, which would add

**F**

__7.___{BC}= m a

_{BC},

making a

_{BC}an intermediate result connecting Equations 6 and 7, just like v

_{B}connects Equations 3 and 6. In principle you can go through the same steps, starting with circles and boxes, but if you need multiple intermediate results you're probably better off using different-colored pens to keep the different parallelograms separate. In practice, you will find that, with experience, you can intuitively identify the equations you will need.

Suggestion: Use the problem-solving sheets to fully work out Examples 1 and 2.